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My physics textbook says two electric field lines never intersect. Their explanation runs somewhat like this:

If two field lines crossed, there would be two different directions to the electric field at the intersection point, which is impossible by definition.

A similar explanation is provided as to why two stream lines never cross.

However, mathematically, two intersecting curves can have the same direction of tangent at their intersection point. For example, consider the $x-$axis and the curve $y=x^3$ is the $xy$ plane. Or any two curves of the form $y=x^{2k+1}$ with $k\in\mathbb N$, plotted for $-1<x<1$.

Is the explanation then wrong, considering these counterexamples? If not, why not? If yes, then what's the correct explanation?

Edit: The answers to this Phys.SE question seem to focus on the very explanation I am having trouble understanding; I don't see how this is a duplicate of that one. I'd rather describe this as a follow-up question to that one.

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Ankoganit
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    Possible duplicate of Why can two (or more) electric field lines never cross? – anna v Apr 03 '17 at 05:14
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    If you draw what you are describing, you are drawing a source at x=0 . Sources are charges in experimental observations and as modeled by classical electrodynamics. electric field lines theoreticlly cross on the point where a point charge is. – anna v Apr 03 '17 at 05:17
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    This is really a mathematical question, related to the existence and uniqueness theorem for ODEs. The answer depends on how pathological you allow your situations to be. – knzhou Apr 03 '17 at 05:25
  • In a real physical setup, this wouldn't be allowed. By allowing field lines to merge into one, you increase the local electric flux density to infinity. – knzhou Apr 03 '17 at 05:26
  • @knzhou I'm interested to know how the existence and uniqueness theorem for ODEs solves my problem; mind elaborating? And FWIW, I don't mind pathological situations. :) – Ankoganit Apr 03 '17 at 06:09

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Electric field lines are a visual construct to understand electric fields. By definition, the density of field lines indicates the magnitude of electric field at a point and the direction of the field lines indicate the direction of the electric field.

If two field lines intersect, then what would be the direction at the point of intersection? Therefore, electric field lines cannot intersect.

Yashas
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  • Then why don't we find the net of the two intersecting lines to get the E. Field .? – Mitchell Apr 03 '17 at 05:43
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    If you had two electric field lines due to two sources at a point, their vector sum would indeed give you the direction of the electric field (and the density would give the magnitude). However, the field line at a location in space by definition accounts for the individual sources. The field line at a location represents the direction of force on a positive charge. – Yashas Apr 03 '17 at 05:53
  • "If two field lines intersect, then what would be the direction at the point of intersection?" Sorry, I don't see how this is a problem. :( As I mentioned in the question, the direction can be well-defined in cases of some intersections. Thanks for your interest. – Ankoganit Apr 03 '17 at 06:12
  • There can be only one field line for a point in space. The field line gives the direction of electric field. You cannot have electric field which has two directions. – Yashas Apr 03 '17 at 06:16
  • I agree that we can't have electric field with two directions; but why not two field lines with one direction? – Ankoganit Apr 03 '17 at 06:28
  • As I said earlier, you can add two field lines up vectorially. The issue is with the definition. The field line is supposed to represent the direction in which the force acts and the density of field lines must give the magnitude of electric field at the point. A point in space can have one and only one field line. Check Field lines. – Yashas Apr 03 '17 at 06:30